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Properties

Label 16.12.2126314463...0625.1
Degree $16$
Signature $[12, 2]$
Discriminant $5^{8}\cdot 13^{4}\cdot 29^{6}\cdot 179^{2}$
Root discriminant $28.71$
Ramified primes $5, 13, 29, 179$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1177

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, -205, 0, 8856, 0, -10315, 0, 5111, 0, -1455, 0, 256, 0, -25, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 25*x^14 + 256*x^12 - 1455*x^10 + 5111*x^8 - 10315*x^6 + 8856*x^4 - 205*x^2 + 1)
 
gp: K = bnfinit(x^16 - 25*x^14 + 256*x^12 - 1455*x^10 + 5111*x^8 - 10315*x^6 + 8856*x^4 - 205*x^2 + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 25 x^{14} + 256 x^{12} - 1455 x^{10} + 5111 x^{8} - 10315 x^{6} + 8856 x^{4} - 205 x^{2} + 1 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 2]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(212631446319650906640625=5^{8}\cdot 13^{4}\cdot 29^{6}\cdot 179^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $28.71$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $5, 13, 29, 179$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{10} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} + \frac{3}{10} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a + \frac{1}{10}$, $\frac{1}{10} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{5} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{2}{5} a - \frac{1}{2}$, $\frac{1}{10} a^{10} - \frac{1}{5} a^{6} - \frac{1}{2} a^{5} - \frac{2}{5} a^{2} - \frac{1}{2}$, $\frac{1}{10} a^{11} - \frac{1}{5} a^{7} - \frac{1}{2} a^{6} - \frac{2}{5} a^{3} - \frac{1}{2} a$, $\frac{1}{50} a^{12} + \frac{1}{25} a^{10} - \frac{1}{50} a^{8} + \frac{8}{25} a^{6} - \frac{1}{2} a^{5} + \frac{9}{50} a^{4} - \frac{1}{2} a^{3} - \frac{13}{50} a^{2} - \frac{1}{2} a + \frac{21}{50}$, $\frac{1}{50} a^{13} + \frac{1}{25} a^{11} - \frac{1}{50} a^{9} + \frac{8}{25} a^{7} - \frac{1}{2} a^{6} + \frac{9}{50} a^{5} - \frac{1}{2} a^{4} - \frac{13}{50} a^{3} - \frac{1}{2} a^{2} + \frac{21}{50} a$, $\frac{1}{3935950} a^{14} + \frac{4477}{3935950} a^{12} + \frac{80542}{1967975} a^{10} - \frac{19717}{1967975} a^{8} - \frac{1668511}{3935950} a^{6} + \frac{1766837}{3935950} a^{4} - \frac{1}{2} a^{3} + \frac{34478}{1967975} a^{2} + \frac{49259}{157438}$, $\frac{1}{3935950} a^{15} + \frac{4477}{3935950} a^{13} + \frac{80542}{1967975} a^{11} - \frac{19717}{1967975} a^{9} - \frac{1668511}{3935950} a^{7} + \frac{1766837}{3935950} a^{5} - \frac{1}{2} a^{4} + \frac{34478}{1967975} a^{3} + \frac{49259}{157438} a$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $13$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 982887.177328 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1177:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 76 conjugacy class representatives for t16n1177 are not computed
Character table for t16n1177 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.725.1, 8.6.94086875.1, 8.8.2576088125.1, 8.6.461119774375.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 16 siblings: data not computed
Degree 32 siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$13$13.4.2.2$x^{4} - 13 x^{2} + 338$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
13.4.2.2$x^{4} - 13 x^{2} + 338$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
$29$29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.4.3.3$x^{4} + 58$$4$$1$$3$$C_4$$[\ ]_{4}$
29.4.3.3$x^{4} + 58$$4$$1$$3$$C_4$$[\ ]_{4}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
$179$$\Q_{179}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{179}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{179}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{179}$$x + 3$$1$$1$$0$Trivial$[\ ]$
179.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
179.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
179.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
179.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
179.2.1.2$x^{2} + 537$$2$$1$$1$$C_2$$[\ ]_{2}$
179.2.1.2$x^{2} + 537$$2$$1$$1$$C_2$$[\ ]_{2}$